Now for the cruel (and possibly wrong) comments and corrections. I think you should say あさひや と いう 本屋, or possibly 本屋 の あさひや. (Not 100% sure on that second one.) Also note that combining kanji and hiragana in a single word (except for endings) can be confusing. Better to stick to hiragana when you don't know all the kanji in a word, usually.

When you say にさん こども の 本 to mean "a few books" I think you might mean to say こども の 本 にさん さつ (satsu is the counter for books). I think it would be more natural to either say exactly how many (if it's only two or three) or not mention the number (saying 本 を かいました would not necessarily imply just one).

When you say 「おむずびまてまて」 と を 読んで見る, what is the "と" for? Just a typo?

とても ゆっくり 読みます does, literally, mean "I read very slowly", but in context it ends up appearing to mean "I will read them very slowly." You could probably say とても ゆっくり しか 読めません, which would mean "I can only read [them] very slowly."

Instead of 私 の たんご say 私 の しっている たんご, because, I think, たんご really means individual vocabulary words more than the sum of all the words you know. 小さい means "small". You should say 少ない (sukunai) which means "few". Also, the "yo" added at the end of the sentence sounds a little off.

I'm not sure that れい に は is a proper construction (although I'm not sure it's wrong either). I would say たとえば, which means "for example." Also, you could say にばんめ の ページ (the number two page) but にページめ (the second page) would be better.

... How long can these comments be?

柴をかり means to gather firewood or brushwood, but I had to look that up. Also, ひる means noon more than lunchtime, although I guess either would do.

When you say と いみで おもいます, I think の いみ だ と おもいます would be more correct. で is usually used as a kind of conjunction.

Here's how I would translate it (no guarantee of accuracy):

"Early one morning, the old man when to the mountain to gather firewood. He worked away gathering wood, and it soon became midday. The old man took out his only omusubi, sat down on the root of a tree, and got ready to eat."

The second sentence could be literally translated "While he was working, it became noon." 食べようとしました could be more literally translated as "He made to eat" or "He tried to eat", but I think my version carries the implied meaning better.

Even the cruel comments are welcome, as long as they're constructively cruel.

When you say 「おむずびまてまて」 と を 読んで見る, what is the "と" for? Just a typo?I could save face and say yes, but honestly, I think I'm a tad screwed up on the rules involving と for things that would "normally" be in quotes.

Good start, and it certainly kept me occupied for a while.Thanks. It's kept me busy too. That second post took me two hours! Hopefully, as I get more practice in, it'll get easier.

I think I'm a tad screwed up on the rules involving と for things that would "normally" be in quotes.

Well, the first good rule of thumb is that you can't put two particles together most of the time. と is, I believe, normally used when you are literally quoting something someone said or wrote. It can also be used when naming things (as in this book) in a phrase like 「おむすびまてまて」と呼びます (It is called "Omusubi Mate Mate"), but it would only be used with 読む when describing how a particular phrase or character is 'read', for example, when the pronunciation was not obvious. When talking about the title of a book you are reading, you can just say 「おむすびまてまて」を読んでみました。

When you want to say what something is called you say ＿＿と呼びます (__to yobimasu). Yobu is also the verb "to call" in the sense of calling someone to come over or to get their attention. The phrase can also be used to identify yourself: コリンと呼びます。 "I'm called Colin."

By the way, if you wanted to ask me how is that said, you should have asked 「呼びます」は なん と 読みます か. (Literally "What does 'yobimasu' read as.") Um, I think.